86.5.16 problem 16

Internal problem ID [23130]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5a at page 74
Problem number : 16
Date solved : Thursday, October 02, 2025 at 09:23:11 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-8 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)-8*y(x) = 0; 
ic:=[y(0) = 2, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 2 \cosh \left (2 \sqrt {2}\, x \right )+\frac {\sqrt {2}\, \sinh \left (2 \sqrt {2}\, x \right )}{4} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 46

ode=D[y[x],{x,2}]-8*y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} e^{-2 \sqrt {2} x} \left (\left (8+\sqrt {2}\right ) e^{4 \sqrt {2} x}+8-\sqrt {2}\right ) \end{align*}

✓ Sympy. Time used: 0.058 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {\sqrt {2}}{8} + 1\right ) e^{2 \sqrt {2} x} + \left (1 - \frac {\sqrt {2}}{8}\right ) e^{- 2 \sqrt {2} x} \]